3D angular velocity

Now consider the rotation of a 3D rigid body. Recall from Section 3.3 that Euler's rotation theorem implies that every 3D rotation can be described as a rotation $ \theta $ about an axis $ v = (v_1, v_2, v_3)$ though the origin. As the orientation of the body changes over a short period of time $ \Delta t$, imagine the axis that corresponds to the change in rotation. In the case of the merry-go-round, the axis would be $ v = (0,1,0)$. More generally, $ v$ could be any unit vector.

The 3D angular velocity is therefore expressed as a 3D vector:

$\displaystyle (\omega_x,\omega_y,\omega_z),$ (8.15)

which can be imagined as taking the original $ \omega $ from the 2D case and multiplying it by the vector $ v$. This weights the components according to the coordinate axes. Thus, the components could be considered as $ \omega_x = \omega v_1$, $ \omega_y = \omega v_2$, and $ \omega_z = \omega v_3$. The $ \omega_x$, $ \omega_y$, and $ \omega_z$ components also correspond to the rotation rate in terms of pitch, roll, and yaw, respectively. We avoided these representations in Section 3.3 due to noncommutativity and kinematic singularities; however, it turns out that for velocities these problems do not exist [308]. Thus, we can avoid quaternions at this stage.

Steven M LaValle 2020-01-06